4.5 Basic probabilityIB Maths: Analysis and Approaches HL: Revision notes

A trial is one performance of an experiment, such as rolling a dice once. Each possible result is an outcome. The set of all possible outcomes is the sample space $U$, and an event $A$ is a set of outcomes, such as 'the score is even'.

When two dice are rolled, write outcomes as ordered pairs: (red 1, blue 2) and (red 2, blue 1) are different outcomes, so the sample space has $6\times6 = 36$ equally likely outcomes.

When all outcomes are equally likely, $P(A) = \frac{n(A)}{n(U)},$ where $n(A)$ is the number of outcomes in $A$ and $n(U)$ the number in the sample space. Every probability satisfies $0 \le P(A) \le 1$, and the probabilities of all outcomes add to 1.

Example: tickets 1 to 30; the multiples of 4 are 4, 8, …, 28, so $P = \frac{7}{30}$.

The complement $A'$ is the event 'not $A$'. Exactly one of $A$ and $A'$ happens, so $P(A) + P(A') = 1, \qquad P(A') = 1 - P(A).$ Use it whenever 'not' or 'at least' makes direct counting long: $P(\text{not prime}) = 1 - \frac{10}{30} = \frac{2}{3}$.

The same idea gives unknown probabilities: if red has probability 0.28 and blue is twice yellow, then $0.28 + 2p + p = 1$.

When outcomes are not equally likely, or the probability is unknown, estimate it from an experiment: $\text{relative frequency} = \frac{\text{number of times the event occurs}}{\text{number of trials}}.$ The more trials, the more reliable the estimate, so combine samples when they come from the same source: $\frac{14+27}{400+600} = 0.041$.

A relative frequency that differs slightly from a theoretical probability does not prove the theory wrong; results vary by chance.

If an event has probability $p$ and there are $n$ trials, the expected number of occurrences is $np.$ For example, 128 students each with probability 0.1 of absence gives an expected $12.8$ absences. The expected number does not have to be a whole number, and it is a long-run average, not a guarantee.